Is there a way to tile only a part of a texture?
This is not possible the way you describe it. What you would usually do is either one of: 


If the rest of the eye is only a single color, you can rescale the UVs for these parts and even overlap them. (In this example you could also cut off most of the white in the texture)

If the rest still needs more than a single color, but less details, and you want to use tiling, then you should use two different textures.